Is it correct to start a sentence with "with" in English?
Solution 1:
There's nothing grammatically wrong with that construction. In fact it's quite common. It's an example of a dependent clause. You can use any subordinating conjunctions in this way.
However, I will agree that it sounds a bit odd for other reasons. The development of the economy speaks of the development almost as if it was a single event. Personally, I'd write it like this:
With the developing economy, living standards improved.